We have the function f(x) = (3/2) x – 9/2 and we have to find the inverse function.

Let y = f(x) = (3/2) x – 9/2

=> y = (3/2) x – 9/2

=> 2y = 3x - 9

=> 2y + 9 = 3x

=> x = 2y/3 + 9/3

=> x = 2y/3 + 3

interchange x and y

=> y = 2x/3 + 3

**Therefore the inverse of f(x) or f^-1(x) = 2x/3 + 3.**

To determine the inverse function, we'll have to prove that f(x) is bijective. We'll re-write f(x) factorizing by (3/2)

f(x) = (3/2)(x - 3)

Since x - 3 is a linear function, that is bijective, then f(x) is bijective too.

We'll put f(x) = y

y = (3/2)(x - 3)

We'll change y by x:

x = (3/2)(y - 3)

We'll have to determine y:

We'll divide by 3/2:

2x/3 = y - 3

We'll isolate y to the right side and then we'll use symmetric property. We'll add 3 both sides:

y = 2x/3 + 3

The inverse function is:

**f^-1(x) = 2x/3 + 3**